Trees in the Reiser4 Filesystem, Part I
One way of organizing information is to put it into trees. When we organize information in a computer, we typically sort it into piles called nodes, with a name (pointer) for each pile. Some of the nodes contain pointers, and we can look through the nodes to find those pointers to (usually other) nodes.
We are interested particularly in how to organize so we actually can find things when we search for them. A tree is an organizational structure that has some useful properties for that purpose. We define a tree as follows:
A tree is a set of nodes organized into a root node and zero or more additional sets of nodes, called subtrees.
Each of the subtrees is a tree.
No node in the tree points to the root node, and exactly one pointer from a node in the tree points to each non-root node in the tree.
The root node has a pointer to each of its subtrees, that is, a pointer to the root node of the subtree.
Note that a single node with no pointers is a tree, because it is the root node. Also, a tree can be a linear tree of nodes without branches.
It is interesting to argue about whether finite should be a part of the definition of trees. There are many ways of defining trees, and which is the best definition depends on your purpose. Professor Donald Knuth supplies several definitions of tree. As his primary definition of tree he even supplies one which has no pointers, edges or lines in the definition, only sets of nodes.
Knuth defines trees as being finite sets of nodes. There are papers on inifinte trees on the Internet. I think it more appropriate to consider finite an additional qualifier on trees, rather than bundling finite into the definition. However, I personally only deal with finite trees in my research.
Edge is a term often used in tree definitions. A pointer is unidirectional, meaning you can follow it from the node that has it to the node it points to, but you cannot follow it back from the node it points to, to the node that has it. An edge, however, is bidirectional, meaning you can follow it in both directions.
Here are three alternative tree definitions, which are interesting in how they are mathematically equivalent to each other. They are not equivalent to the definition I supplied, because edges are not equivalent to pointers. For all three of these definitions, let there be no more than one edge connecting the same two nodes:
a set of vertices (aka points) connected by edges (aka lines) for which the number of edges is one less than the number of vertices;
a set of vertices connected by edges that have no cycles (a cycle is a path from a vertex to itself);
or a set of vertices connected by edges for which there is exactly one path connecting any two vertices.
These three alternative definitions do not have a unique root in their tree, and such trees are called free trees.
The definition I supplied is a definition of a rooted tree, not a free tree. It also has no cycles, it has one less pointer than it has nodes, and there is exactly one path from the root to any node.
Consider the purposes for which you might want to use a graph and those for which you might want to use a tree. In a tree there is exactly one path from the root to each node in the tree, and a tree has the minimum number of pointers sufficient to connect all the nodes. This makes it a simple and efficient structure. Trees are useful when efficiency with minimal complexity is desired and when there is no need to reach a node by more than one route.
Reiser4 has both graphs and trees, with trees used when the filesystem chooses the organization (in what we call the storage layer, which tries to be simple and efficient) and graphs when the user chooses the organization (in the semantic layer, which tries to be expressive so that the user can do whatever he or she wants).
We assign a key to everything stored in the tree. We find things by their keys, and using them gives us additional flexibility in how we sort things. If the keys are small, we have a compact means of specifying enough information to find the thing. It also limits what information we can use for finding things.
This limit restricts the key's usefulness, and so we have a storage layer, which finds things by keys, and a semantic layer, which has a rich naming system (described in Part II of this article). The storage layer chooses keys for things solely to organize storage in a way that improves performance, and the semantic layer understands names that have meaning to users. As you read, you might want to think about whether this is a useful separation that allows the freedom to add improvements that aid performance in the storage layer, while escaping paying a price for the side effects of those improvements on the flexible naming objectives of the semantic layer.
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